The diffeomorphic excision of closed local compacta from infinite-dimensional Hilbert manifolds

C ^OMPOSITIO M ^ATHEMATICA

J AMES E. W EST

The diffeomorphic excision of closed local compacta from infinite-dimensional Hilbert manifolds

Compositio Mathematica, tome 21, n

^o

3 (1969), p. 271-291

<http://www.numdam.org/item?id=CM_1969__21_3_271_0>

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271

The diffeomorphic excision of closed local compacta from infinite-dimensional Hilbert manifolds

by

James E. West 1

Wolters-Noordhoff Publishing

Printed in the Netherlands

If M is an infinite-dimensional

manifold,

which open, dense submanifolds of it are

homeomorphic

^or

diffeomorphic

^{to it}

by

functions

arbitrarily

^{close to the}

identity?

^{R. D.}

^Anderson,

^David

W.

Henderson,

and the author

together

have shown

[2]

^{that if M}

is a metrizable manifold modelled on a

separable,

infinite-dimen- sional Fréchet space, then each open, dense submanifold N of M with the

property

^{that for} each open set U of

M,

^{U and U n N}

have the same

homotopy type

^is

homeomorphic

^{to M}

by

^{a homeo-}

morphism

which may be

required

^{to be the}

identity

^on any closed subset of M

lying

^{in N} and may be limited

by

any open ^cover of M.

(A

^function

f

^{from a} subset X of M into M is said to be limited

by

the open ^cover G of M if the collection

((r, f(x)}|x e X)

^refines

G.)

Such submanifolds include the

complements

^{of all}

^closed, locally compact

subsets of

M,

but the method of

proof

^{used cannot}

readily

be

adapted

^to

give diffeomorphisms

^{when M is a} differentiable

manifold,

^{as it involves}

homeomorphisms

between Fréchet spaces which ^are not

diffeomorphic.

^The

principal

tools used in

[2]

may be traced

conceptually

^{from the}

proof

^{due to V. L.}

^{Klee, Jr.,} [5],

that a

separable,

infinite-dimensional Hilbert space is homeo-

morphic

^{to the}

complement

of each of its

compacta.

^{In 1966,}

Cz.

Bessaga [3] produced

^a differentiable version of Klee’s theorem in the

special

^{case of a}

single point,

^so it seemed natural to the author to

try proving

^a differentiable version of

[2]

^for

comple-

ments of

closed, locally compact

subsets of differentiable manifolds

on

separable,

infinite-dimensional Hilbert spaces

(real).

^{The anal-}

ogy is

complete,

^{as the}

following

^statement of the theorem of this paper shows:

Il

M is a metrizable

CP-manifold (1 p

^Ç

~)

modelled on

separable, in f inite-dimensional

Hilbert spaces, ^X is a

closed, locally compact

^subset

of ^M,

^{U is an} open subset

of

^{M con-}

taining ^X,

^{and G is an} open ^cover

of ^M,

then there is a

CP-diffeo- morphisms of

^{M onto}

^MBX

which is the

identity off

^U and is limited

1 Supported ⁱⁿ part by ^{National Science} Foundation grant ^{GP 7952X.}

by

^{G. The}

proof

^is

elementary

^{in the sense that it}

requires only

^the

Inverse Function

Theorem,

differentiable

partitions

^of

unity,

^and

Bessaga’s result,

^which

requires nothing

^more

sophisticated

^{in its}

proof.

After

completing

^{most of the work on} this paper, the author ^was

apprised by

^{R. D.} Anderson that

(a) by

^minor modifications of

[1],

it is

possible

^{to show that a number of}

topological linear

spaces which possess Schauder bases ^are

C~-diffeomorphic

^{to the}

eomple-

ments of each of their

compacta

^and

(b )

Peter Renz of the Univer-

sity

^of

Washington

^has

recently

^obtained

by

^a different method the result that a metrizable manifold modelled on a

separable,

infinite-dimensional Hilbert space is

diffeomorphic

^{to the}

comple-

ment of each of its closed local

compacta.

The author has also learned from David Henderson that

quite recently

^D.

Burghelea,

N.

Kuiper,

and N. Moulis have proven results

implying

^{that each}

two open subsets of a

separable,

infinite-demensional Hilbert space which have the ^same

homotopy type

^are

C°°-diffeomorphic.

Throughout

^this

paper, H

will denote ^a

separable,

^infinite-

dimensional

(real)

Hilbert space, and

differentiability

^{will be}

taken in the sense of Fréchet. The term "manifold" will denote a

manifold without

boundary.

^{The term}

"C°°-partition

^of

unity

^on

H" is taken to mean a collection S of C°°-functions s from H into

[0,1]

^{and a}

locally

^finite open ^cover

{Us}

^{of H such that}

for each x in

H;

^{S is} said to be subordinate to an open ^cover G of H if

{s-1((0, ~)) 1 s

^E

S}

refines G. Given _any open ^cover G of

H,

^{there is a}

C’-partition

^of

unity

subordinate to G

(see [6];

p.

30).

The

proof

of Theorem 1 is broken into a sequence of ⁸

lemmas,

several of which are not new but are included for purposes of

completeness

and reference.

LEMMA 1.

Il

^{X is a}

closed, locally compact

^subset

of ^H,

^{there is a}

complete,

orthonormal basis

{en}~n=1 for

^{H with the}

property

^that

i f Hl

^{is the}

closed,

linear span

of {e2n-1}~n=1

^and

Pl

^{is the}

(orthogonal) projection of

^{H onto}

Hl,

^then

P1 is

^a

homeomorphism

^{on each}

compact

^subset

of

^X.

PROOF: Because X is a

separable, locally compact,

metric space, it is

possible

^{to find a} collection

{Xi}~i=1

^of

^compacta

^{of X for}

which each

X is

contained in the interior

(relative

^to

X)

^{of its}

successor and X is the union of the

X i’s.

^Let

{zn}~n=1

^{be a}

complete,

orthonormal basis for

H,

^{and for} each i and n let

By

^the

compactness

^{of the}

X i’s, {ai,n}~n=1

^{converges to zero for}

each i. Let

{n(i)}~i=1

^{be a}

subsequence

^{of the}

positive integers

^such

that for

each i,

_ai,n(i)

~ 1/i,

and observe that

if ? ~ i,

^then aj,n(i) ai,n(i). ^Let

{Ak}~k=1

^{be an} infinite collection of

pairwise disjoint

infinite subsets of the

positive integers

such that if each

Ak

is indexed

by

^the

positive integers

^{in the} natural order and is denoted

by {m(k, p)}~p=1,

then for each k and _p,

m(k, p) &#x3E;

^2k+p·

Let,

^{for each}

k,

xk ⁼

03A3~p=1 2-p/2zn(m(k,p)).

Since for each

k,

^the

point

Yk ⁼

2-1 2zn(m(k,1))-zn(m(k,2))

^is

orthogonal

^{to each} xj and the

yk’s

^{are all}

orthogonal,

^{there is a}

complete,

orthonormal basis

{en}~n=1

for H such that for each _{n, e2n = xn} · Such a basis will

suffice,

^for if x and y ^are in X and

P1 ( x ) = P1(y),

^then x2013y is in the

closed,

linear span of the

xk’s. Thus,

Hence,

but since

it is true that

therefore,

for each p.

However,

since there is an i for which both x

and y

^are

in

X i ,

^and

m(k’, p) &#x3E; ^2k’+p

for each p,

if p &#x3E; i,

^then

Thus,

^for

each p &#x3E; i,

This shows that for each

k, (x-y, xk)

^{- 0}

^{and, thus,}

^{that x = y}

and

Pl

^is one-to-one on

X,

^which _proves ^{the lemma.}

LEMMA 2.

Il Hl

^{is a}

^closed,

^linear

subspace of ^H, Pl

^{is the}

projec-

tion

of

^{H onto}

Hl,

^{X is a} closed subset

of ^H,

^and

P1|X

^{is a homeo-}

morphism of

^{X onto a} closed subset

of Hl,

^then

for

any sequence

{fi}~i=1 of C~-diffeomorphisms of

^{H onto}

itself satisfying

^the

four

conditions

below,

^the

uniform

^limit

f of {fi ··· f1}~i=1

^{is a homeomor-}

phism of

^{H onto}

itself

such that

f|P1(X)

⁼

(P1|X)-1

^and

f|HBP1(X)

^{is a}

C~-diffeomorphism.

a) {fi ··· f1}~i=1

^is

uniformly Cauchy.

b) P1fi

⁼

Pl, for

^{all i.}

c) fi+j

^{is the}

identity

outside the open

21-i-neighborhood of

^X

and the

image

^under

fi ··· f1 of

the open

21-i-neighborhood of P1(X), for

^each

i.

d)

^{For each x in}

^X,

^{x =}

limiti~~ fi ··· f1(x).

PROOF: Condition

(a) provides

the existence of

f

^{as defined and}

its

continuity;

^condition

(c)

^{ensures that}

f|HBP1(X)

^{is a C°°-}

diffeomorphism

^{onto its}

image,

^{and condition}

(d)

^{is the statement}

that

f|P1(X)

⁼

(P1|X)-1. Therefore,

^the

only remainin g things

^to

establish are that

f

^is one-to-one, ^that

f(H) = ^H,

^{and that}

f -1

^is

continuous on X. Conditions

(c)

^and

(d) immediately yield

^that

f(H) =

^{H. To see that}

f

^is one-to-one, observe that if there were

two

points,

^{x and} y, of H for which

f(x)

⁼

f(y),

^{then one of}

^them,

say ^x, would have to be in

P1(X),

and the other would have to be in

HBP1(X).

^Condition

(c)

would then

specify

that there is a

positive integer

^{i and an} open ^set

containing fi

^···

f1(y)

^{on which}

fi+j

^{is the}

identity

^for

each i

&#x3E; 1,

which, together

with the fact that

fi

^···

fi

^{is a}

homeomorphism

^of

H,

^{shows that}

f(x)

^~

f(y),

after all. The

continuity

^of

f-1

^at

points

^{of X} is assured

by (b), (c),

^and

(d) together,

^{for if x is in X and U is an} open ^set

containing P1(x),

^{then for an i such that the open}

22-i-neighborhood

^of

P1(x)

lies in

U, (b), (c),

^and

(d) give

that the open ^set

is carried

by fi - - - fi

^{onto an} open ^set V

containing

^x

(where B01

is the open unit ball in

H1

^{about the}

origin

^and

B2

^{is the} open unit ball in the

orthogonal complement

^of

Hl

^{about the}

origin). ^Now,

conditions

(b)

^and

(c) guarantee

^that

fi+j

carries Tl onto itself for

each i ^~

^{1. Therefore}

f-1(V)

^is contained in

U,

^and

f-1

^{is conti-}

nuous.

LEMMA 3.

Il

^{Y is a}

separable,

metric space, G is ^an open ^cover

of

Y,

and X is a

closed, locally compact

^subset

of ^Y,

^{then there is a star-}

finite

open ^cover

{Ui}~i=1 of

^{Y and a cover}

{Xi}~i=1 of

^X

by compacta of

X such that

for

^each

^i, Xi

is contained in

Ui,

^and

{Ui}~i=1 refines

^G.

PROOF: This may be done

easily by embedding

^Y in the Hilbert

cube, taking

^{the closure of the}

image

^of

^Y,

^and

using

^the

compact-

ness of the Hilbert cube after the fashion of Theorem 1 of

[2]

^and

Theorem 1 of

[4].

LEMMA 4.

If X is a ^closed, locally compact

^subset

of

^{H and}

Hl

^{is a}

closed,

^linear

subspace of

^H

for

^which

(a) XBH1

^is

compact

^and

(b) i f Pl

^{is the}

projection of

^{H onto}

Hl,

^then

P1|X is a homeomorphism,

then

for

any

positive

real number 8 and open ^set W

of Hl containing P1(X)

^{there is a}

pair, f

^and g,

of homeomorphisms of

^{H onto}

itself satisfying

^the

following:

1) g is

^the

identity off

the intersection

of P-11(W)

with the open

e-neighborhood of

^{X and moves no}

point

^{more than e;}

2 ) g|(HBX) is a C~-diffeomorphism

^onto

g(HBX);

3) f

^{is a}

C~-diffeomorphism

^{and is the}

identity off P-11(W);

4) Pif

⁼

P1g

⁼

Pl, and f

⁼

fP1+I-P1,

^{where I is the}

identity,

and

5) fg|X

⁼

PIIX.

PROOF: Let

H2

^{be the}

orthogonal complement

^of

Hl,

^let

P2

^be

the

projection

^{of H onto}

H2 ( P2

⁼

I - Pl ), ^and,

^for

convenience,

assume

that 03B5 ~

^{1. Let a be a}

non-increasing

C°°-function from the real numbers into

[0,1]

for which

a-1(0)

^D

[1, ~)

^and

a-1(1)

^D

( - oo, 0].

^{Let b =}

sup {|a’(t)||t real},

and observe that from the Mean Value

Theorem, b &#x3E;

^{1. For each}

positive

^{number c,}

define g, ^to be the function from

H1

^{to the} real numbers such that

gc(x )

⁼

a(~x~2/c2).

Each gc is ^a

C°°-function,

and for each x in

H,

~g’c (x)~

^ç

2b/c.

^Let cl ^E

(0, 03B5/4)

^{be small}

enough

that the closed

2cl-neighborhood

ⁱⁿ

P1(X)

^of

P1(XBH1)

^is

compact.

^Let

GI

^{be an}

open ^cover of

Hl refining {W, H1BP1(X)}

^{for which}

(1)

^{U E}

G, implies

^that

sup

{~P2(P1|X)-1(x)-P2(P1|X)-1(y)~|x,

y ^E

P1(X)

ⁿ

U}

^~

c1/4b,

and

(2)

each element of

Gl

^has diameter less than cl . Let yl be ^a function from

GI

^into

H2

^{such that}

y1(U)

^{is in}

and is the

origin

otherwise. Let

S,

^{be a}

C°°-partition

^of

unity

^on

Hl

subordinate to

G1,

^{and let u be a} function from

Si

^into

G,

^{such that}

u(s) D s-1 «0, ~))

for each s in

Sl. Set Il

^{to be the function from}

Hl

^into

H2

^defined

by f1(m) = 03A3s~S1 s(x)y(u(s)),

^{and let}

fi

⁼

I+f1P1.

It is immediate that

il

^{is a}

C°°-diffeomorphism

^{of H}

onto itself and is the

identity

^off

P-11(W).

^Let

f

⁼

f-11

⁼

I -f-, Pl.

Let

Ki

be the closure of

{x

^E

P1(X)|f1(x) ~ x},

^{and note that}

by (2)

and the choice of ci,

Kl

^is

compact.

^Let

If

Bi

is the closed unit ball of

Hi

centered about the

origin,

^then

The

homeomorphism

g will be constructed ^{as a} uniform limit of

C°°-diffeomorphisms

^{of H onto} itself defined below.

Let {fi}~i=1

^{be a} sequence of

C°°-diffeomorphisms

^{of H onto itself}

with

fi

^{as above}

satisfying

^the

following

five conditions:

1)

^{there is a set}

{ci}~i=1

^of

positive ^numbers,

^with cl ^as

above,

such that for

each i, 2ci+1

ci 2-2i _s;

2)

^if

i &#x3E; 1,

^then

fi(x)

⁼

x+gci-1 (x-fi-1

^···

f1P1(x)fiP1(x),

where

f,

^{is a} C°°-function from

Hl

^into

(Ci-l/4b )B2

for which the open

c,-neighborhood

^of

P1(X)

^{and W both}

contain f-1i(H2B{0});

3)

^for

each i, pi=(P1|X)-1-fi···f1|P1(X)

^carries

P1(X)

into

(ci/4b)B2;

4) ciB2+fi ··· f1(H1)

^C

fi ’ . f1(2-iB2+H1),

^and

5)

^if y is in

Hl

and x is in

P1(X), then ~y-x~

ci

implies that IIP2fi-l

^...

fl(y)-P2fi-l ... f1(x)~

^2-2i03B5.

Such a

sequence {fi}~i=1 ^exists,

for the conditions are

arranged

to

provide

^an easy inductive construction as follows:

If a

collection {fi}ni=1, n ~

^1,

of diffeomorphisms

^is

given satisfying

conditions

(1)-(5),

^let

c’n+1

^{be a}

positive

^{number so} small that for y in

Hl

^{and x in}

P1(X) with ~y-x~ c’n+1,

Let

ti, ..., and t.

^{be in}

(0,1)

such that if 0

~ x ~ ^ti,

^then

|1-gci(x) ^~ 2{b/ci)~x~, for each i

^n.

(This

^can be done because

gé, ^{(o )}

^{is the zero}

functional,

^{and hence}

Let

and let

dn+1

^E

(0, cn+1)

^{be so} small that for x

and y

ⁱⁿ

P1(X)

^with

Now,

^let

Gn+l

^{be an} open ^cover of

H1

which refines

{W, H1BP1(X)}

^{and is of mesh less than}

dn+1;

let yn+1 :

Gn+1 ~ H2

be ^a function such that

yn+1(U)

^{is in}

and is the

origin otherwise;

^let

Sn+1

^{be a}

C°°-partition

^of

unity

^on

H1

subordinate to

Gn+l’

^{and let} un+1 :

Sn+1 ~ Gn+l

^{be a function}

such that for s in

Sn+1, u.+, (s) D s-1 «0, ~)).

^Let

for each x in

H1,

^{and define}

At this

point,

it will be shown that

fn+1

^{is a}

C°°-diffeomorphism

of H onto itself. The

proof

^{is a standard}

argument involving

^the

Inverse Function

Theorem,

the Mean Value

Theorem,

^{and the}

Banach Contraction

Principle (for explicit

statements of these theorems and for

proofs,

^see pages ¹¹ and 12 of

[6]).

The Inverse Function Theorem

implies

^{that in order to show that}

fn+1 is

^{a C°°-}

diffeomorphism

^{of H onto}

itself,

^it is sufficient to show that

fn+1

is one-to-one, ^that

fn+1(H)

^-

^H,

and that for each x in

H, fn+1(x)

is a linear

homeomorphism

^{of H onto} itself. In order to show that

fn+1

^is one-to-one and carries H onto

itself,

it suffices to show that for each _y in

H,

^the function oy of

H2

into itself defined

by

^the

formula

oy(x)

⁼

P2(y)-gcn(x-P2fn ··· f1P1(y))fn+1P1(y)

^{has a}

unique fixed-point,

^since

if xy

^{is a fixed}

point

of oy, then

Thus,

and

In order to show that oy ^{has a}

unique

^fixed

point,

^{the Banach}

Contraction

Principle

^{asserts that it suffices to find a k in}

(0,1)

such that for each x and x’ in

H2, ~oy(x)-oy(x’)~ ~ k~x-x’~.

The Mean Value Theorem shows that if k is a uniform bound on the

norm of the derivative of oy, then this

happens.

^{For each y in}

^H,

the constant k may be taken ^to

be 1,

for if y is in H and x is in

H2,

where ^et." denotes the scalar

multiplication

^{of the} linear functional and the element of H. Since

/.+,

^{is a} one-to-one map of H ^onto itself.

To

complete

^the verification that

fn+1

^{is a}

C°°-diffeomorphism

^of

H onto

itself,

^there

only

^{remains to} show that for each x in

H,

f’n+1(x)

^{is a linear}

homeomorphism

^{of H onto itself.}

By

^{the Closed}

Graph Theorem,

^{this is}

equivalent

^to

showing

^{that for}

each x,

fn+1(x)

^is one-to-one and carries H onto itself. For each x and y in

H,

Because

both fn+1 P1(x) and f’n+1 (P1(x)) (P1(y))

^{lie in}

^H2,

^{the kernel}

of f’n+1(x)

^{must also lie in}

^{H2. However, H2}

^{is an invariant}

^subspace

of

f’n+1(x),

^and

f’n+1(x)|H2 is

^{a linear}

homeomorphism

^of

H2

^onto

itself,

since for y in

H2 ,

and

Thus, f’n+1(x)

^is one-to-one, ^{and since}

fn+1

^{is a}

C’-diffeomorphism

^{of H onto itself.}

The

collection {fi}n+1i=1

satisfies conditions

(1)-(5)

^rather

easily.

Conditions

(1)

^and

(5)

^{are met}

explicitly by

^the choice of cn+1, and condition

(2)

is satisfied

by

^the construction of

fn+1,

^{the fact}

^that, by (3), PnPl(X)

^{lies in}

(cn/4b)B2’

and the fact that no element of

Gn+1 meeting P1(X)

^contains

points

^of

H1BW

^or farther than cn+1 from

P1(X).

Condition

(3)

^{is met} because if x is in

P1(X),

^then

and, therefore,

by

^{the choice of}

dn+1

^.

In order to see that

(4)

^is

satisfied,

observe that for each x in

H1, fn+1|(H2+x) is

^a

C~-diffeomorphism

^of

H2+x

^{onto itself.}

Hence, fn+1 ··· f1(2-n-1t1 ··· tnB02+x)

^{is an open}

neighborhood

ⁱⁿ

H2+x

of

fn+1 ··· f1(x).

^The

argument

below shows that it contains

cn+1B02+fn+1 ··· f1(x).

^If y is in

h2+x, y ~ x,

^and

then

This is true because

by

^{the choice of}

ti.

^{A similar}

argument yields

^{the other}

part

^{of the}

inequality. Thus, if ~y-x~ = 2-n-1t1 ··· tn,

^{then since}

(for y in H2+x),

^{an induction shows that}

The set of all

such y

ⁱⁿ

H2+x

^{is the}

boundary

ⁱⁿ

H2+x

^of

so its

image

^under

fn+1 ··· fi

^{must be the}

boundary

^of

in

H2+x,

^{which must therefore contain}

Since conditions

(1)-(5)

^are

satisfied,

^{an induction shows the}

existence of ^an infinite _sequence

{fi}~i=1

^of

C°°-diffeomorphisms

^of

H onto itself which ^meets all five of the conditions. These five conditions

imply

^{the four} conditions ^{of Lemma 2.} Conditions

(1)

and

(2) imply (a); (2)

^{and the} definition ^of

il imply (b),

^and

(1)

and

(3) imply (d).

^{To show that}

(c) ^holds,

^{let i}

and i

^be

positive integers,

^and

let y

be in H. If

ii+i(y)

^{e y,}

^{then, by (2),}

~P1(y)-P1(x)~

^{ci+, for some x in X}

^{and, by (5),}

Also,

^since

furthermore, by (3),

Combining

^these

inequalities gives

Therefore, fi+i

^{is the}

identity

outside the open

2-iB-neighborhood

of X.

Also,

^since

if fi+j(y) ~ y, then y

^{is in}

ci+j-1B02+fi+j-1···f1(H1),

(4) yields that y

^{is in}

fi+j ··· f1(2-iB02+H1),

^so

Now, (2)

shows that

~P1(y)-P1(x)~

ci., for ^{some x} in

X; thus,

an d (c ) holds.

By

^{Lemma 2,} the uniform limit h

of {fi ··· f1}~i=1

^{is a homeo-}

morphism

^{of H onto} itself which is ^a

C°°-diffeomorphism

^off

P1(X)

and which ^on

P1(X)

^{agrees with}

(P1|X)-1.

^{Let g =}

f 1 h-1.

^Since

each

f four 1

&#x3E; 1 is the

identity

^{off the} open

a-neighborhood

^of

^X,

g is ^the

identity

^{off the}

e-neighborhood

^of

^X,

and because

g ^{moves no}

point

^{as much as E. It is easy to}

verify

^that

f

and g are the desired

homeomorphisms

^{of H.}

LEMMA 5.

Il

^{X is a}

compact

^subset

o f

^H

lying

in the open ^set

U,

then there is a real-valued

function f of

^{H into}

[0,1] of

class C°° such that X ⁼

f-1(0)

^{and HBU C}

f-1(1).

PROOF: This is an easy

generalization

from the well-known result in the case that H is finite-dimensional.

(Or

^see

[8], chapter V,

^{for a} discussion of

carriers.)

LEMMA 6

(Bessaga ). Il Hl

^is any

closed, in f inite-dimensional,

linear

subspace of

^{the real} Hilbert space

E,

then there is ^a

C°°-di f f eo- morphism

^h

of EB{0}

^{onto E} which is the

identity off

the unit ball

of

E centered at the

origin

and has the

property

^that

(I-h)(EB{0})

is contained in

Hl.

A

proof

of this lemma _{may be} found in

[3].

LEMMA 7 .

Il

^{X is a}

^closed, locally compact

^subset

of

^the

^closed,

linear

subspace Hl of

^{H and}

i f H2

^{is a}

^closed, in f inite-dimensional,

linear

subspace of

^the

orthogonal complement Hi of Hl,

^then

for

^any

open ^set U

containing X,

^{there is a}

C~-diffeomorphism

^h

of ^HBX

onto H which is the

identity off

U and has the

property

^that

(I - h) (HBX)

is contained in

H2 .

PROOF: The internal direct sum of two

closed, orthogonal,

linear

subspaces, Hi

^and

H,,

^{of H will be denoted}

by Hi+Hj;

the

symbol "C+D"

^{will continue to denote the set of all sums of}

pairs

^of

elements,

^one from C and the other from

D,

for any sub- sets, ^{C and}

D,

^{of H. As}

before, Pi

will denote the

projection

^{of H}

onto the

closed,

^linear

subspace Hi,

^and

Bi

will denote the closed unit ball of

Hi

^{centered at the}

origin.

^Let

H4

^{be a}

one-dimensional,

linear

subspace

^of

H2,

^{and let}

H3

^{be its}

orthogonal complement

ⁱⁿ

H2.

^{Let e be an} element of

H4

^{of norm one,} and let T be a linear

isomorphism

^{of H onto}

H1+H3

^{for which}

PIT

⁼

Pl.

There is a

C°°-diffeomorphism f

^{of H onto} itself such that

2) f-1(H1+H3) ~ (H,+H 3) == X+H3 = T(X+H~1), 3) P4f|y+H3

^{is constant for each} y in

H1,

^and

4)

^for

each y

ⁱⁿ

(H1+H3)BT(U), ~P2f(y)~ ^~

^{1. The}

folllowing

three

paragraphs provide

^a construction of such ^a function.

For each x in

X,

^let

Vx

^{be a}

relatively

open ^set in

Hl containing

x for which there is

a d.,

ⁱⁿ

(0,1)

^{such that}

Vx+dxB3

^{is contained}

in

T(U). By

^Lemma 3, ^there exist open ^covers,

{Ui}~i=1

^and

{Wi}~i=1,

^of

H1

^{and a cover}

{Xi}~i=1

^{of X}

by compact

subsets of X such

that,

^for

each i, Xi

^C

Ui

^C

Ui C Wi

and such that

{Wi}~i=1

^is

a star-finite refinement of

{H1BX}

^u

{Vx}x~X.

Let,

^for

each i,

ai be a C°°-function from

H1

^into

[0,1]

^{for which}

a-1i(0) ~ (H1BWi)

^and

a-1i(1) ~ Ui,

^unless

Ui

^{n X =}

0,

^{in which}

case

ai(Hl) = {0}. (If Ui () X =F 0,

ai may be obtained from a C°°-

partition

^of

unity

^of

Hl

subordinate to

{Wi, H1BUi) by summing

all elements which vanish ^{on a}

neighborhood

^of

H1BWi.)

^{For each}

i such that

Ui ~ X ~ 0,

^let

x(i)

^{be an} element of X for which

Vx(i)

^D

W i , and,

^{for each}

i,

^let

Define a:

H1 ~ [0, ~) by a(x)

^-

03A0~i=1 (1+(1/di)ai(x)), where 03A0

denotes real

multiplication.

Let g : H - ^H be defined

by

Now,

^if y is in

(H1+H3)BT(U)

^and

P1 (y )

^{is in}

then

~P3g(y)~

^{~ 1} because there is an i for which

P1(y)

^{is in}

Ui and, so, ~P3(y)~ ~ di. (Thus,

By

^{Lemma 5,} for each i there is a

C°°-function bi

^from

H1

^into

[0,1]

^{such that}

b-1i(0)

^-

X and b-1i(1) ~ H1BUi,

^{with the}

proviso

that if

Xi

⁼

^0,

^then

bi(Hl)

⁼

{1}.

^{Let b :}

H1 ~ [0,1]

^{be defined}

by b(x)

⁼

03A0~i=1 bi(x),

^{and note that}

b-1(1) ~ HlBA ; furthermore,

b-1(0) =

X. The function

f

may ^now be defined

by

By

^{Lemma 6,} there exists a

C~-diffeomorphism p

^of

H~1B{0}

onto

Hi

which is the

identity

off the unit ball of

Hi

^{centered at}

the

origin

^{and has the}

property

^that

(1 -p )(Ht) C H3.

^Let

h ⁼

T-1f-1(P1+p(I-P1))f(TBHBX).

^{This is} the desired diffeo-

morphism

^of

^HBX

^{onto H.}

LEMMA

8. If X is a ^closed, locally compact

^subset

o f ^H,

^{U is an open}

subset

o f

^H

containing ^X,

and E is a

positive

^real

^number,

^{then there}

is a

Coo-diffeomorphism o f ^HBX

^{onto H} which is the

identity o f f

^U

and moves no

point

^{more than B.}

PROOF:

By

^{Lemma 3, there is a} star-finite _open cover

{Vi}~i=1 refining {HBX, U}

^{and a cover}

{Xi}~i=1

^{of X}

by compact

^{subsets of}

X which have the

properties

^that

(1) Vi

^{n X is}

compact,

^{for each}

i,

^and

(2)

^for

^{each i,} X C Vi . By

^{Lemma 1, there are three}

^closed,

linear

subspaces, Hl, H2,

^and

H3,

^{of H such that each two are}

orthogonal,

^{H =}

H1+H2+H3, H2

^and

H3

^are

infinite-dimensional,

and

P1

^{is a}

homeomorphism

^{on each}

compact

subset of X.

Let

Al

⁼

{Vi1}, ^{where il}

is the least

integer i

^{for which}

V

n

X ~ 0, and, assuming A1, ···, An-1

^to be defined with

Ar denoting

^the union of all elements of

Aj, let An = {Vi| Vi ft ~n-1k=1Ak, Vi

ⁿ

^{X ~} 0,

^{and either}

Vi

ⁿ

A*n-1 ~ ~

or i is the least

integer

^for

which Vi

^{satisfies the first two}

conditions}.

^Let

Yn = ~{Xi|Vi~An}.

The collection

{A*n}~n=1

^{has the}

property that |n-m| &#x3E;

¹

implies

that

A*m ~ A*n = ^0,

^{and each}

Yn

^is

compact.

Let d be the function from the set of

pairs

of subsets of H to the real numbers defined

by d(A, B)

^{= inf}

{~a-b~|a

^e

^A,

^{b e}

^B},

and for

each n,

^let

d?n-1

^{be a}

positive

number less than

Now,

^for

each n,

^set

Z2n-l

^{to be the closed}

2d2n-l-neighborhood

of

Y2n-l in ^X,

^{and note that each}

Z2n-1

^is

^compact

and its open

1 2d2n-1-neighborhood

^{lies in}

A*2n-1.

^Consider

{P1(Z2n-1)}~n=1.

Because

Plis

^an open map and is ^a

homeomorphism

^on

Z2n-3 U Z2n-l U Z2n+l’

^{for each n, there is a} collection

{W*2n-1}~n=1

of _open sets in

H1

^{for which}

P1(Z2n-1)

^C

W2n-l C P1(A*2n-1)

^and

W2n-3 n W2n-l == W2n-l n W2n+l

⁼

^0,

^{for each n;}

furthermore,

there is a collection

{W2n}~n=1

^{of open sets of}

^Hl

^{for which}

P1(Y2n) ~ W2n ~ P1(A*2n)

^and

W 2n -2 n W 2n

⁼

W 2n rl W2n+2 = ^0.

for each n.

By

^Lemma 4, ^there

is,

^for

eacb n,

^a

pair, f2n-l

^and

g2n-l’ ^of

homeomorphisms

^{of H onto} itself such that

f2n-l is

^{a C°°-}

diffeomorphism

^{of H which is the}

identity

^off

P-11(W2n-1)

^{and is}

a translation of each

hyperplane parallel

^to

H2 + H3

^into

^itself,

g2n-l is

^a

C°°-diffeomorphism

^on

HBZ2n-1,

^{is the}

identity

^{on the}

complement

^of

P-11(W2n-1)

^{and on the}

complement

^{of the} open

2d2n-lweighborhood

^of

^Z2n-l’

^{and moves no}

^point

^{more than}

1 2d2n-1, f2n-1g2n-1|Z2n-1 = P1|Z2n-1,

^and

P1g2n-1 = P1.

Let

d’2n-1

^e

(0, d2n-1)

^{be small}

enough

^that

(a)

the open

d2n-z-

neighborhood

^of

f2n-1g2n-1(Y2n-1)

ⁱⁿ

f2n-1g2n-1(X)

^{lies in}

f2n-1g2n-1(Z2n-1) = Pl(Z2n-l)’ (b)

^thé open

4d2n-lweighborhood

of

f2n-1g2n-1(Y2n-j)

^{lies in}

f2n-1(A*2n-j)

^and

f2n-1g2n-1(A*2n-j),

^for

j

^{= 0, 1, and 2, and}

(c)

^the open

4d’2n-1-neighborhood

^of

f2n-1g2n-1(Z2n-1)

^lies

P-11(W2n-1). By

^{Lemma 7, there}

is,

^{for each}

n, ^a

C°°-diffeomorphism h2n-l

^of

HBP1(Y2n-1)

onto H which is the

identity

^{off the} open

d2n-i-neighborhood

^of

P1(Y2n-1)

^and

has the

property

^that

(P1+P3)h2n-1

⁼

P1+P3.

Each

h2n-l

^{is the}

identity

^on

f2n-1g2n-1(Y2n-2BZ2n-1)

^and

f2n-1g2n-1(Y2nBZ2n-1)

^and carries the _open

d’2n-1-neighbor-

hoods of

f2n-1g2n-1(Y2n-2BY2n-1)

^and

f2n-1g2n-1(Y2nBY2n-1)

ⁱⁿ

HBP1(Y2n-1)

^into

f2n-1(A*2n-2)

^and

f2n-1(A*2n), respectively.

^This

is because if z is in H and

then, by (b),

^the open

3d’2n-1-neighborhood

of z lies in

f 2n-1(A2 -2)-

Since

z-h2n-1(z) is

ⁱⁿ

H2,

^if

h2n-1(z) ~ z, ^then,

^as

h2n-l

^is

the

identity

^off the open

d’2n-1-neighborhood

^of

P1(Y2n-1),

~z-h2n-1(z)~ 2d2n-1 and, hence, h2n-l(Z)

^{is in}

f2n-1(A*2n-2).

^The

same

argument gives

^that

h’2n-1

^carries the open

d’2n-1-neighborhood

Of f2n-1g2n-1(Y2nBY2n-1)

^into

f2n-l (A:n).

Let

F2n-1 = (f-12n-1h2n-1f2n-1g2n-1)|HBY2n-1. By (b )

^and

(c ),

each

F2n-l

^{is the}

identity

^off the intersection of

P-11(W2n-1)

^with

A*2n-1

^{and is a}

homeomorphism

^of

HBY2n-1

^{onto H which is a}

C°°-diffeomorphism

^off

Z2n-l.

^{Define F :}

HBU~n=1Y2n-1 ~ H by F(x)

⁼

limitn~~ F2n-1 ’ ’ ’ F1(x),

^for each x in

HBU~n=1Y2n-1.

Since

{A*2n-1}~n=1

^{is a}

locally

finite collection of sets

(by

^{virtue of}

of the fact that

{Vi}~i=1

^is

star-finite)

^{and the sets}

A*2n-1

^are

pairwise disjoint,

^{F is a}

homeomorphism

^{which on}

HBU~n=1Z2n-1

is a

C°°-diffeomorphism.

^{Because each}

A*n

^{lies in}

^U,

^{F is}

the identity

off U.

Consider,

now, the collection of sets

Each of these sets

lies, except

^{for a} subset with

compact closure,

in

H1+H2,

^and

(P1+P2)|Z2n is

^a

homeomorphism

^of

Z2n

^into

Hl+H2.

^{This statement} may ^be verified ^as follows: Because

Y2nBY2n-1 ~ Y2n+l) is a closed, locally compact

^{subset of}

HBU~n=1 Y2n-l’ Z2n

^{is a}

^closed, locally compact

subset of H.

Because

F2n-1

^{is the}

identity

^off

A*2n-1,

^{for each n, these functions}

commute, ^and

By

the condition on the sets

{W2m-1}~m=1,

^{each of the functions}

h2n+1, f2n+l’

^and g2n+l ^commutes with all of the functions

h2n-i, f2n-l’

and g2n-l. Therefore

and since

lies in

H1

^and

h2n+lh2n-l(H1)

^{lies in}

H1+H2,

lies in

H1+H2. ^However, Z2n

is the union of this set with

which is

compact;

so,

Z2nB(H1+H2)

^is

compact.

^{To see that}

(P1+P2)BZ2n is

^a

homeomorphism

^of

Z2n

^into

H1+H2,

^observe

that,

from the definitions of the functions

involved,

and

is a

homeomorphism

^of

Z2n

^onto

Y2nB(Y2n-1 ~ Y2n+l).

^Because

P1 is, by

^{Lemma 1, a}

homeomorphism

^on

Y2nB(Y2n-1 ~ Y2n+l)’

Plis

^a

homeomorphism

^on

Z2n. Therefore,

^since

(P1+P2)|Z2n is

^{also a}

homeomorphism.

For

each n,

^let

d2n E (0,

^min

{d’2n-1, d’2n+1})

^{be small}

enough

^that

the open

d2n-neighborhood

^of

is contained in

f2n+1g2n+1f2n-1g2n-1(A*2n).

^This

requirement

^{is suffi-}

cient to

guarantee

that the open

d2n-neighborhood

^of

Z2n

^{also lies}

in

f2n+1g2n+1f2n-1g2n-1(A*2n).

This follows because if ^a

point x

^{of H}

is within

d2n

^{of a}

point

y of

Z2n,

then in the case

that y

^{is in}

the

spécifie

^choice

of d2n

shows that x is in

f2n+1g2n+1f2n-1g2n-1(A*2n),

while in the case that _y is in

f2n-1f2n+1F(Z2n±1B(Y2n±1

^U

Y2nI2))’

then, by

^condition

(c)

^on the choice of the set

{d’2m-1}~m=1,

^both

x and _y lie in

P-11(W2n±1),

^so

h2n~1(y)

- y, and from the fact that

~h-12n±1(y)-y~ d’2n±1,

^{it is true}

^that ~x-h-12n±1(y)~ 4d’2n±1,

which,

^from conditions

(b)

^and

(c)

^{on the choice of}

d’2n±1

^{and the}

fact that

f2n":fl

^and g2n~1 ^{are the}

identity

^off

P-11(W2n~1), gives

^that

x is ln

f2n+1g2n+1f2n-1g2n-1(A*2n).

Lemma 4

gives

^a

collection {f2n, g2n}~n=1

^of

pairs

of homeomor-

phisms

^{of H onto itself such that each}

f 2n

^{is a}

C°°-diffeomorphism

of

H, f2n(x+y)

⁼

f2n(x)+y

for each x in H and _y in

H3, (P1+P2)f2n = Pl+P2 = (P1+P2)g2n, f 2n

and g2n ^are the

identity

off

P-11(W2n), g2n is

^the

identity

off the open

2d2n-neighborhood

^of

Z2n’

g2n ^{moves no}

point

^{more than}

d2n,

^and

Lemma 7

gives

^a collection

{h2n}~n=1

of functions such that each

h2n

^{is a}

C’-diffeomorphism

^of

HBf2ng2n(Z2n)

^onto

^H, h2n(x)-x

^is

in

H3

for each x in

HBf2ng2n(Z2n),

^for

^{each n,}

^{and each}

h2.

^{is the}

identity

off the intersection of

Pll(W2n)

^{with the}

image

^under

/2n

^of the open

d2n-neighborhood

^of

Z2n

and with the open

d2n- neighborhood

^of

H1+H2.

^Let

and note that

F2n

^{is the}

identity

^off

(This

^{is true because}

f2n,

g2n’ and

h2n

^{are the}

identity

^off

Pll(W2n)’

g2n is the

identity

^{off the open}

d2n-neighborhood

^of

Z2n,

^{which lies}

in

f2n+1g2n+1f2n-1g2n-1(A*2n),

^and

h2n

^{is the}

identity

^{off the}

image

under

f 2n

^{of the} open

d2n-neighborhood

^of

Z2n’

^which

together yield

^that

F2n

^{is the}

identity

^off

which is

P-11(W2n)

^~

g2n+1g2n-1(A*2n).

^Since g2n:i:l ^{is the}

identity

^off

A*2n±1, F2n

^{is the}

identity

^off

P-11(W2n) ~ (A*2n-1 ~ A*2n ~ A*2n+1).)

Thus,

^the

F2n’s

^{are the}

identity

^{off a} collection of

pairwise disjoint

open sets, ^the closures of which form a

locally

finite collection of sets in

H,

^so

G (x )

^-

limitn~~ F2n ··· F2(x)

^{is a} C°°-diffeomor-

phism

^of

HBF(XBU~n=1Y2n-1)

^{onto H} which is the

identity

^{off U.}

Let h ⁼

G(F|HBX).

^{Now h is a}

C~-diffeomorphism

^of

^HBX

^onto

H which is the

identity

^{off U. To}

verify that ~h(x)-x~

^{03B5 for}

each x in

HBX,

observe that if x is in

A*2n-1

ⁿ

Pll(W2n-l)’

^then

and since

h2n-1(y)-y is

ⁱⁿ

H2,

^{for all} y in the domain of

h2n-l’

^and

f2n-1(x+P2(y))

^-

f2n-1(x)+P2(y)

for each x

and y

ⁱⁿ

^H,

~F(x)-x~ d2n-1+2d’2n-1. If,

^{on the other}

hand, x

^{is not} in any

A*2n-1 ~ P-11(W2n-1),

^then

F(x)

^{= x.}

^Now,

^if y is in

then

and since

f2n(x)-x, g2n(x)-x,

^and

h2n(x)-x

all lie in

H3,

^{for each}

x in the domains of these

functions,

therefore,

^{for each x in}

HBX,

for some m and _n, which is less than e.

THEOREM 1. Eeach matrizable

C-"-manifold

^{modelled on}

separable, in f inite-dimensional

Hilbert spaces is

CP-diffeomorphic

^{to the com-}

plement of

^each

of

^its

^closed, locally compact subsets;

^{moreover, the}

di f f eomorphism

may be

required

^{to be the}

identity o f f

any open ^set

containing

^the

locally compact

^{set in}

question

^and may be limited

by

any open ^cover

of

^the

manifold.

PROOF: Let

M, X, U,

^{and G be the}

manifold, locally compact

set, open set, _{and open} cover in

question.

^Because the diffeomor-

phism

^to be constructed may be defined ^on each

component

^{of M}

separately,

it may be assumed ^{that lll} is connected

and, hence,

separable

^{and modelled on the}

separable,

inf inite-dimension al Hilbert space H.

It suffices to prove ^the

following

^statement

(Statement A):

If

Vo, ..-, V n

^are open subsets of ^H and

Xo, ..., X n

^are

locally compact

subsets of

V0, ···,

^and

Vn, respectively,

^{which are rela-}

tively

^{closed in}

Uni=0 Vi,

then there is a

C°°-diffeomorphism

^of

(Uni=0ViBV0

^onto

Uni=0 V

ⁱ which is the

identity

^off

V0BX0

^and

carries

XiBX0

^into

Vi,

for each i ⁼ 1, ..., ^{n. This is true because} then

using

^the definition of M and Lemma 3, ^{there are}

collections, {Vi}~i=1, {Wi}~i=1,

^and

{Xi}~i=1,

of subsets of M such that

{Vi}~i=1

is an open ^cover of M which is a star-finite refinement of G and is ^a refinement of

{U, MBX},

each element of which is Cp-diffeo-

morphic

^{to an} open subset of ^H

by

^{a function}

f i ,

^each

X

i is a

compact

^{subset of}

X, X

⁼

U~i=1Xi, {Wi}i=1 is

^an open ^cover of

M, and,

^for

each i, Xi ~ W C Wi C Vi. Now,

Statement A

gives

^a

C~-diffeomorphism hl

^of

f1(V1BX1)

^onto

f1(V1)

which is the

identity off f1(W1)

^{and carries}

f1(XiBX1)

ⁿ

Fi)

^into

f1(Wi

^~

V1),

for each i. Let gl ^{be the} natural extension of

f-11h1(f1|V1BX1) to MBX1. Inductively,

for each i &#x3E; 1, ^let

hi

^{be a}

C°°-diffeomorphism

of

onto

which is the

identity

^off

and carries

into

fi(Wk

ⁿ

Vi ),

for each k &#x3E; i. Define gi ^to be the natural exten- sion of

to gi-1 ’ ’ ’

g1(MBUj~iXj). Require

^{that if}

Xi

⁼

0,

^then

hi,

^hence

gi, be the

identity.

Since gi is the

identity except when Vi

^{C U}

and since gi ’ ’ ’

g1(x) ~

gi-1 ···

g1(x) implies x

^{is in}

Tli ,

^{there is a}

well-defined

G’p-diffeomorphism g(x) -== limiti~~

gi ’ ’ ’

gl(x)

^from

MBX

^onto M. The function

g-1

^{is the}

identity

^off

^{U, and,}

^because

{Vi}~i=1

^{is a} refinement of

G, g-1

is limited

by

^G.

In order to prove Statement

A,

^{first note} that Lemma 8

easily

implies

^that

(Statement B)

if U and V ^are two open subset of

H,

^V is contained in

U,

^{and Y is a}

locally compact

^{subset of V}